Ura Midterm Report - " Refining the Lattice Package "

In the previous semester, we considered some fundamental ideas in the theory of group representations over finite fields. Specifically, we studied Norton's Irreducibility Criterion and Parker's Meat-Axe algorithm, two tools for investigating the reducibility of a representation. Given these tools, we wish to consider the more complicated problem of finding all subrepresen-tations of a group representation over a finite field. In our previous report, we said that if G is a group, a group representation ϕ : G → GL(n, F) is reducible if there exists a nontrivial proper subspace of F n that is invariant under the elements of ϕ(G). We also saw that for every proper, nontrivial subspace S invariant under ϕ(G), we could devise two simpler representations from ϕ, one describing the action of ϕ(G) on S and the other describing the action of ϕ(G) on the quotient space F n /S. Thus, one way to find all subrepresentations of a given group representation ϕ is to find all of the subspaces of F n invariant under the elements in ϕ(G). Since the vector spaces in question are over finite fields, one might na¨ıvely try to solve this problem by repeatedly applying the generators of ϕ(G) to each of the vectors in F n. Because many of the representations we want to study involve vector spaces of dimension 100 or more, this approach is computationally infeasible. A more tractable approach, implemented by Lux et al. is to deduce a set of spaces P such that each member is invariant under ϕ(G) and contains maximally a unique space invariant under ϕ(G). The spaces in P can then be used to deduce all of the other spaces invariant under ϕ(G). Crucially, once we determine P , the theory tells us all other invariant subspaces can be expressed as the sums of spaces in P. These calculations, which amount to determining subspace relationships, are considerably less time consuming than determining the images of vectors under the elements of ϕ(G). Before we discuss this procedure in more detail, it may be interesting to consider two applications of this algorithm. In coding theory, many encoding